Stochastic ranking for constrained evolutionary optimization

Abstract

Abstract-Penalty functions are often used in constrained optimization. However, it is very difficult to strike the right balance between objective and penalty functions. This paper introduces a novel approach to balance objective and penalty functions stochastically, i.e., stochastic ranking, and presents a new view on penalty function methods in terms of the dominance of penalty and objective functions. Some of the pitfalls of naive penalty methods are discussed in these terms. The new ranking method is tested using a ( ) evolution strategy on 13 benchmark problems. Our results show that suitable ranking alone (i.e., selection), without the introduction of complicated and specialized variation operators, is capable of improving the search performance significantly.

Simple Feasibility Rules and Differential Evolution for Constrained Optimization

“…A comparison of the performance of CHDE with respect to three techniques that are representative of the state-of-the-art in the area: the Homomorphous maps [16], Stochastic Ranking [9] and the Adaptive Segregational Constraint Handling Evolutionary Algorithm (ASCHEA) [18] are presented in Tables 3, 4 Table 4. Comparison of our approach (CHDE) with respect to the Stochastic Ranking (SR)…”

“…To evaluate the performance of the proposed approach we used the 13 test functions described in [9]. The test functions chosen contain characteristics that are representative of what can be considered "difficult" global optimization problems for an evolutionary algorithm.…”

“…The test functions chosen contain characteristics that are representative of what can be considered "difficult" global optimization problems for an evolutionary algorithm. Their expressions can be found in [9] To get a measure of the difficulty of solving each of these problems, a ρ metric (as suggested by Koziel and Michalewicz [16]) was computed using the following expression: ρ = |F |/|S|, where |F | is the number of feasible solutions and |S| is the total number of solutions randomly generated. In this work, S = 1, 000, 000 random solutions.…”

“…The main motivation of this work was to analyze if the use of the selection criteria that we successfully adopted in evolution strategies would also work with differential evolution. This is an important issue to us, because it has been hypothesized in the past that evolution strategies are a very powerful search engine for constrained optimization when dealing with real numbers [9]. However, no such studies exist for differential evolution nor other related heuristics that operate on real numbers (as evolution strategies).…”

“…The approach showed a good performance in problems with only inequality constraints but presented problems when dealing with equality constraints. Moreover, only two test functions (out of seven used to test the approach) are included in the well-known benchmark for constrained optimization proposed by Koziel & Michalewicz [16] and enriched by Runarsson & Yao [9]. The main drawback of the approach is that it adds two userdefined parameters and that the N T parameter can cause an increase in the number of evaluations of the objective function without any user control.…”